# Determining the size of power sets

A very elementary question:

Let $|A|=a$, and $|B|=b$. What's $|\mathcal P(\mathcal P(\mathcal P(A \times B \times \varnothing)))|$?

So how I got stuck on it: I'm not sure what's the number of pairs that a generic $A \times B$ gives, and what's $A \times B \times \varnothing$? Is it $A$ times $B$ times empty set? Sorry, I'm kinda confused on all of this subject and don't have a good source of knowledge (terrible lecturer and no textbook...).

P.S. Is there a good online source, or an eBook or whatever for discrete mathematics?

Thanks in advance for any help, tips etc.

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$A\times B\times \varnothing=\{(a,b,c): \underbrace{a\in A \wedge b\in B\wedge \underbrace{c\in \varnothing}_{\text{Impossible}}}_{\text{Impossible}}\}=\varnothing$ – Git Gud Mar 31 '13 at 21:56
Oh I see. Thanks!!! – ohad Mar 31 '13 at 21:57

Hint: $$A\times B\times\varnothing=\varnothing.$$

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Disregard. Thank you!!! – ohad Mar 31 '13 at 21:57

$$P(\emptyset)=\{\emptyset\}$$

$$P(P(\emptyset))=\{\emptyset,\{\emptyset\}\}$$

$$P(P(P(\emptyset)))=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\},\{\emptyset,\{\emptyset\}\}$$

$$|P(P(P(A\times B\times \emptyset)))|=|P(P(P(\emptyset)))|=4$$

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